Question

Is function composition associative?

Answer 1

Yes

Explanation:

Given composable functions `f`, `g` and `h`

`(f@(g@h))(x)`

`= f((g@h)(x)) = f(g(h(x))) = (f@g)(h(x))`

`= ((f@g)@h)(x)`

So `f@(g@h) = (f@g)@h`

Answer 2

It is, if the following works:

`(f@(g@h))(x) = ((f@g)@h)(x)`

That is, if:

`f(x) = "something"`
`g(h(x)) = (g@h)(x) = "something else"`
`f(g(x)) = (f@g)(x) = "something else again"`
`h(x) = "something else yet again"`

...and you can use these together to satisfy the first expression, then they are associative. Let:

`f(x) = 2x`
`g(x) = x^2`
`h(x) = x^3`

Thus:

`g(h(x)) = g(x^3) = (x^3)^2 = x^6`

`f(g(x)) = g(x^2) = 2(x^2) = 2x^2`

Then:

`(f@(g@h))(x) = f(x^6) = 2(x^6) = color(blue)(2x^6)`

`((f@g)@h)(x) = f(g(x^3)) = f((x^3)^2) = f(x^6) = color(blue)(2x^6)`

Therefore they are associative.